
\magnification = 2200 %\magstep3remains is


      \font\teniscld=cmmi10 at 10 pt
      \font\seveniscld=cmmi10 at 7 pt
      \font\fiveiscld=cmmi10 at 5 pt     
      \font\tensyscld=cmsy10 at 10 pt
      \font\sevensyscld=cmsy10 at 7 pt   
      \font\fivesyscld=cmsy10 at 5 pt    
      \font\tenexscld=cmex10 at 10 pt
      \font\tenbfscld=cmbx10 at 10 pt
      \font\sevenbfscld=cmbx10 at 7 pt    
      \font\fivebfscld=cmbx10 at 5 pt  



      \font\brm=cmr10 scaled \magstep1 
      \font\bbrm=cmr10 scaled \magstep2 
      \font\bbbrm=cmr10 scaled \magstep3 
      \font \bbbbrm=cmr10 scaled \magstep4 
      \font\bbbbbrm=cmr10 scaled \magstep5 
      \font\sf = cmss10 
      \font\chapfont=cmbx10 scaled \magstep2 
      \font\secfont=cmbx10 scaled \magstep1 
      \font\sc= cmcsc10
     \font\cmrviii=cmr10 at 8pt
     \def\abstractfont{\cmrviii}
      
     \font\mathit=cmmi10
      \font\mathitvii=cmmi10 at 7 pt
      \font\mathitv=cmmi10 at 5  pt   
      \font\mathsym=cmsy10 at 10 pt
      \font\mathsymvii=cmsy10 at 7 pt   
      \font\mathsymv=cmsy10 at 5 pt    
      \font\mathex=cmex10 at 10 pt
      \font\mathb=cmbx10
      \font\mathbvii=cmbx7 
      \font\mathbv=cmbx5



%% definitions of control sequences for characters in the typewriter font %%%
%%%% for short phrases this is easier than using a literal construction %%%%
\def\\{\char92{}}          %%%%% backslash %%%%%
\def\lb{\char'173{}}       %%%%% left bracket %%%%%
\def\rb{\char'175{}}       %%%%% right bracket %%%%%
\def\sp{\char32{}}         %%%%% special space symbol %%%%%

\def\beginliteral{
\vskip\baselineskip
\begingroup
\tt
\obeylines
\catcode`\@=0
\parskip=0pt\parindent=0pt
\catcode`\$=12\catcode`\&=12\catcode`\^=12\catcode`\#=12
\catcode`\_=12\catcode`\=12
\def\par{\leavevmode\endgraf}
\catcode`\{=12\catcode`\}=12\catcode`\%=12\catcode`\\=12
}

\def\endliteral{\endgroup}
     
\def\noblackboxes{\overfullrule=0pt}
     
\long\def \proclaim #1. #2\par {
    \smallskip \noindent 
     {\bf  #1. \hskip -5pt} {\sl #2} \par \smallskip}
     
%%%%%%%%%%% macro to put a box around the text %%%%%%%%%%%%%%%%
\def\makebox#1#2#3% vsize, hsize, inserted text
{\hbox{\vrule
       \vbox to  #1{\hrule \vss
                   \hbox to #2{\hss#3\hss}\vss
                   \hrule}\vrule}}
\def\displaytext#1{$$\hbox{#1}$$}


\def\rmarginput#1{
\font \marginfont= cmr7
\def\strutdepth{\dp\strutbox}
\setbox4\vtop
{\hsize 4 pc\marginfont \baselineskip 7.5pt\noindent \raggedright #1\par}
\strut
\vadjust
{\kern -\strutdepth
\vtop to \strutdepth
{\baselineskip\strutdepth\vss\rlap{\hskip\hsize\hskip 1em\box4}\null}}
\kern-22pt}

\def\rmi{\rmarginput}



%%%   RSP Defs

\def\sph{\hbox{\bf S}}
\def\reals{\hbox{\mathb R}}
\def\Cx{\hbox{\bf C}}
\def\Z{\hbox{\bf Z}}
\def\On{\mathop{\hbox{\bf O}}}
\def\SOn{\mathop{\hbox{\bf SO}}}
\def\GLn{\mathop{\hbox{\bf GL}}}
\def\SLn{\mathop{\hbox{\bf SL}}}
\def\Un{\mathop{\hbox{\bf U}}}
\def\SUn{\mathop{\hbox{\bf SU}}}
\def\hsph{\hat{\sph}}
\def\intS{\int_S}
\def \norm|#1|{\left \Vert#1\right\Vert}
\def\enditem{\par}  

\def\qedmark{\hbox{\vrule height 4pt width 3pt}}
\def\qedskip{\vrule height 4pt width 0pt depth 1pc}
\def\qed{\nobreak\quad\nobreak{\qedmark\qedskip}} 
\def\HN{H_{\kern-1pt{}_N}}
\def\Hinfty{H_{\kern-1pt{}_\infty}}
\def\fN{f_{{}_N}}
\def\sphN{\sph_{\kern-1pt{}_N}}
\def\om{\omega}
\def\omN{\om_{\kern-1pt{}_N}}
\def\intSN{\int_{S_N}}
\def\sumSN{\sum_{z \in S_N}}
\def\dsphN{{\hat{\sph}}_{\kern-1pt{}_N}}

\def\Lin{\mathop{\hbox{\bf L}}}

\def\Hess{\mathop{\roman {Hess}}\nolimits}
\def\Spec{\mathop{\roman {Spec}}\nolimits}
\def\hess{\mathop{\roman {hess}}\nolimits}
\def\Div{{\mathop \bold {div}\nolimits}}
\def\Per{{\mathop \bold {Per}\nolimits}}

\def\es{\epsilon}

\def\half{{1 \over 2} }
\def\nr{{1 \over \sqrt{2 \pi}} }
\def\ip{\nr \int _0^ {2 \pi}}
\def \IP<#1,#2>{\left \langle #1,#2\right\rangle}
\def\sp{ \sum_{j=0}^{N-1} }

\def\intr{{\hbox{\subandsup int}}}
\def\extr{{\hbox{\subandsup ext}}}


\def\Comment//#1//{\setbox0\hbox{#1 }}


%%%%%
%% Inserting Graphics %%%

\newbox\imagebox \newdimen\imageheight
\def\image#1#2{%
  \setbox\imagebox=\hbox{\pdfximage{#1}%
    \pdfrefximage\pdflastximage}
  \imageheight=\ht\imagebox
  \divide\imageheight by\mag
  \multiply\imageheight by1000
  \setbox\imagebox=\hbox{\pdfximage height#2\imageheight{#1}%
    \pdfrefximage\pdflastximage}
  \box\imagebox}
 
%  Convert the graphic image to pdf by either opening it in Acrobat
%   or using Photoshop,If the final graphic image file is for example
%  paraboloid.pdf then use
% \image{paraboloid.pdf}{,<scalefactor>}
% to insert the image scaled by scalefactor.



\input amssym.def            % small letters for UNIX,  not: AMSsym.def

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum

\def\bu{$\bullet$\hskip 4pt}
\def\p{\partial}
\def\I{\hbox{} {\rm I\/}}
\def\II{\hbox{} {\rm II\/}}

\def\R{\hbox{\bf R}}
\def\ni{\noindent}
\def \o {\theta}

\long\def \refclaim[#1]#2. #3\par {\advance \parno by1
    \medbreak \noindent 
{\bf \parnum \hskip 6 pt #2.\hskip 6pt}%
\expandafter\edef\csname ref#1\endcsname
{\parnum}\ifproofmode\rmarginsert[\string\ref#1]\fi
{\sl #3} \par \medbreak}




\hsize 7.25 true in 
\vsize 9 true in
\hoffset = -0.20 true in 
\voffset -0.25 true in
\parskip=3pt

\nopagenumbers

\vglue 10pt

\cl{\brm Ruled Surfaces}
\bigskip \ni
  Informally speaking, a {\bf ruled surface} is one that is a union of straight lines  (the rulings). To be more precise, it is a surface that can be represented parametrically in the form:  \hfil\break
  $$x(u,v) = \delta(u) + v*\lambda(u)$$
where $\delta$ is a regular space curve (i.e., $\delta'$ never vanishes) called the {\bf directrix} and $\lambda$ is a smooth curve that does not pass through the origin.  Without loss of generality, we can assume that $|\lambda(t)| = 1$.
For each fixed $u$ we get a line 
$v \mapsto \delta(u) + v*\lambda(u)$ lying in the surface, and these are the 
 rulings. (Some surfaces can be parameterized in the above form in two essentially different ways, and such surfaces are called {\bf doubly-ruled surfaces}.)
 
\ni
A ruled surface is called a {\bf cylinder} if the directrix lies in a plane $P$ and 
$\lambda(u)$ is a constant direction not parallel to $P$, and it is called a 
{\bf cone} if all the rulings pass through a fixed point $V$ (the vertex).

\vfil\eject
\ni
Two more interesting examples are the {\bf Hyperboloid of One Sheet} 
$${x^2\over a^2} + {y^2\over b^2} - {z^2\over c^2} = 1,$$
 which is in fact doubly-ruled, since it can be given parametrically 
 by:
 $$x^+(u,v) = a(\cos(u) - v \sin(u)), b(\sin(u) + v \cos(u),c v)$$ 
and 
 $$x^-(u,v) = a(\cos(u) + v \sin(u)), b(\sin(u) - v \cos(u),-c v),$$
 and the {\bf Hyperbolic Paraboloi}d:
 $$(x,y,z) = (a \,u, b\, v, c \,u \,v) = a(u,0,0) + v(0,b,c\,u).$$

\ni
Another interesting ruled surface is the {\bf Helicoi}d minimal surface:
$$ \eqalign{
F(u,v) &= b( \cosh(v) \cos(u), \cosh(v) \sin(u), v) \cr
          &= (0,0,bv) + b\cosh(v)(\cos(u),\sin(u),0)\cr
}
 $$
 \vfil\eject
\ni
A ruled surface is called a (generalized) {\bf right conoid} if its rulings are parallel to some plane, $P$, and all pass through a line $L$ that is orthogonal to $P$.
{\it The\/} {\bf Right Conoid} is given by taking $P$ to be the $xy$-plane and $L$ the $z$-axis. It is given by:
$$ F(u,v) =  (v\cos(u), v \sin(u), 2 \sin(u)).$$

\ni
This surface has at $(\sin(u) = \pm 1, v=0)$ two pinch point singularities. Famous for such a
singularity is the {\bf  Whitney Umbrella}, another right conoid with rulings parallel to the
x-y-plane:
$$ F(u,v)  = ( u\cdot v, u,  v\cdot v) .$$

\ni
In both cases one can emphasize the visualization of the singularity by embedding these
surfaces into families of ruled surfaces. We deform the {\bf Right Conoid to a Helicoid} so
that the stable pinch point singularity disappears, at the final moment through an unstable
singularity:
$$ \eqalign{
F(u,v) =  \left(\matrix{v\cos(u) ,  v \sin(u) ,  2aa \sin(u) +(1-aa)u}
 \right).
}$$

\ni
And we deform the {\bf Whitney Umbrella} so that it closes at the top and thereby slowly
forms another pinch point singularity:
$$ \eqalign{
F(u,v)  = \left(\matrix{ u\cdot (aa\cdot  v + (1-aa)\sin(\pi v)) \cr
            \  u   \cr      
             aa\cdot  v^2 - (1-aa)\cos(\pi v)}
 \right) .
}$$

\ni
R.S.P. \& H.K.

\bye